In the past recent years, the Navanlinna characteristic of \( f(z+c) \) , the value distribution theory for difference polynomials, the Nevanlinna theory for the difference operator and most importantly, the difference analogue of the lemmas on the logarithmic derivative had been established (see [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}). For these theories, since the derivative is a difference counterpart of a function, hence there has been recent study of whether the derivative \( f^{\prime } \)  of a entire (meromorphic) function \( f \)  can be replaced by the difference \( \Delta _c(f)=f(z+c)-f(z) \)  in the above mentioned results. Number of researches have been done with difference operator (see[10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [7]}, [18]})
